The proof for this is based on the cosine rule for triangles. Let A and B be the vectors you are considering. Now in vector terms we know A + C = B (following from head to tail of both vectors) which means that C = B - A and this means the length is the length of B - A (which you can use pythagoras rule for in n-dimensions).

Now your cosine rule is C^2 = A^2 + B^2 - 2ABcos(theta). You know how to calculate lengths of all the vectors (using Pythagoras') so know collect the terms together and see what you get.

The length is known for arbitrary finite n through Pythagoras' theorem and the proof using lengths works in any dimension for R^n: it's a very simple proof since you only care about lengths of the triangle and it's very easy to understand (length is an invariant concept)